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Brannagan was spotted as a seven-year-old by Liverpool scouts whilst playing for Cadishead in 2004. Brannagan is a childhood Manchester United fan, however he turned down an offer from them to make the move to Liverpool, after he and his father noted the Reds' impressive coaching schemes and training facilities. He began life at the academy with the Under 9s, and has progressed over the years to the Under 18s- for whom he made his debut in the 2011-12 season. This came on 24 March 2012 in a 4-3 loss to West Brom, with Brannagan entering as a second half substitute. On 1 August 2012, Brannagan netted a brace against Rochdale in an Under 18s 2012-13 pre season match. He became an integral part of the U18s in the 2012-13 season, racking up 18 appearances, netting one goal and he also made his debut for the U21s during the season. He progressed to become a more permanent fixture in the U21s for the 2013-14 season. He netted his first goal for the Liverpool second string on 23 September 2013, with a close range effort to seal the Reds' 2-0 win over Man City. He enjoyed a breakthrough season with the U21s, racking up 19 appearances and netting 4 goals.

Brannagan also became involved in first team affairs at Melwood, and even made the bench for the first team's FA Cup clash against Oldham on 5 January 2014. On 14 May 2014, he appeared as a second half substitute in Liverpool's post-season friendly win over Shamrock Rovers. Ahead of the 2014-15 season, Brannagan adopted shirt number 32, with new signing Lazar Markovic requesting Brannagan's number 50 shirt.

On 17 September 2015, Brannagan finally made his first team debut, coming on as an 80th minute substitute in a 1-1 draw at Bordeaux. On 26 October 2015, Brannagan confirmed via his Twitter account that he had signed a new contract with the club. The next day, Jurgen Klopp said that Brannagan "brings everything you need for a midfield player". He made his first start for the club a day later, playing 65 minutes in a 1-0 win at home to Bournemouth. Brannagan put in a solid performance, and received a standing ovation from the Kop.

Brannagan mad his next start on 8 January 2016, when he played the full 90 minutes in a 2-2 FA Cup draw to Exeter. Brannagan continued to start in the cup, playing the whole of the replay against Exeter at Anfield. On 1 May 2016, Brannagan made his Premier League debut, coming on as a late substitute in a 3-1 defeat at Swansea. Two weeks later, he made his first Premier League start, playing the whole of a 1-1 draw at West Brom.

In the summer of 2016, Brannagan gave up his number 32 shirt for new signing Joel Matip, and was instead handed the number 25.

Brannagan is a versatile footballer who has previously enjoyed a spell at right back with the Academy, however he now tends to operate as an attacking midfielder behind the striker or out wide as a winger.